Small optimal Margulis numbers force upper volume bounds

Abstract

If $\lambda$ is a positive real number strictly less than $\log 3$, there is a positive number $V_{\lambda }$ such that every orientable hyperbolic 3-manifold of volume greater than $V_{\lambda }$ admits $\lambda$ as a Margulis number. If $\lambda <(\log 3)/2$, such a $V_{\lambda }$ can be specified explicitly, and is bounded above by $$\lambda (6+\frac{880}{\log 3-2\lambda }\log \frac{1}{\log 3-2\lambda }),$$ where $\log$ denotes the natural logarithm. These results imply that for $\lambda <\log 3$, an orientable hyperbolic 3-manifold that does not have $\lambda$ as a Margulis number has a rank-2 subgroup of bounded index in its fundamental group, and in particular has a fundamental group of bounded rank. Again, the bounds in these corollaries can be made explicit if $\lambda <(\log 3)/2$.

29 pages

References in corpus (3)

• Tameness of hyperbolic 3-manifolds
• Minimum-volume hyperbolic 3-manifolds
• Presentation length and Simon's conjecture